Friction always opposes the motion; static friction opposes the imposed force - until the limit is reached. Then the coefficient will change.

If there was no friction the wall would constrain the block. Since there is friction, it must be taken into account.

Update: imagine a pressure sensor on the wall; when there is no friction it would give one reading (force applied / cross section area of the block). With friction the force appearing at the wall will be reduced by the frictional force opposing the motion).

Friction always opposes the motion; static friction opposes the imposed force - until the limit is reached. Then the coefficient will change.

If there was no friction the wall would constrain the block. Since there is friction, it must be taken into account.

Update: imagine a pressure sensor on the wall; when there is no friction it would give one reading (force applied / cross section area of the block). With friction the force appearing at the wall will be reduced by the frictional force opposing the motion).

I'm not so sure. If the block cannot even attempt to slide then it looks more like the wall takes all of the force. Perhaps someone with more knowledge than myself could answer this.

For real systems (i.e., not perfectly rigid) both forces will exist. The origin of friction are microscopic interactions at the interface between the materials ... these interactions exist independent of any lateral force ... thus the friction willl be apparent when the block is pushed.

Thus the presence of the wall is a secondary condition - the force at the wall will be the force of the push minus the force of the static friction.

Now consider what happens when the applied force exceeds the limits of the static coefficient of friction!

Thanks, UltrafastPED and Drakkith.
In the case of the block, since the limiting value of the static friction is 5*10*0.2=10N, the reaction from the wall should be 190N.
I wonder if someone in the forum can conduct an experiment to confirm/demonstrate this phenomenon.

Dear DaleSpam,
Does that mean that we can not assume that the static friction will be at the limiting value?
Thus, as the friction can take any value, the reaction from the wall is indeterminate?
Can we not rig up an experiment using different surfaces with different coefficients of friction, and measure the reaction from the wall, keeping the surface horizontal and the wall vertical?

Dear DaleSpam,
Does that mean that we can not assume that the static friction will be at the limiting value?
Thus, as the friction can take any value, the reaction from the wall is indeterminate?

Yes and yes.

Can we not rig up an experiment using different surfaces with different coefficients of friction, and measure the reaction from the wall, keeping the surface horizontal and the wall vertical?

We can, but it won't help. Consider that if the force pushing the block towards the wall is high enough, we could remove the floor and the block would still be fixed in place against the wall; that is, the characteristics of the floor including surface and coefficient of friction are irrelevant to the outcome under some conditions.

If you carefully review the Wikipedia article on statically indeterminate systems you will find this statement:
"Considerations in the material properties and compatibility in deformations are taken to solve statically indeterminate systems or structures."

This is why I took into consideration the microscopic origins of friction, and also discarded the "rigid body" idealization. And yes, a real experiment could be performed.

Agreed, a real experiment could be performed. But the scenario being described here does not pin down a set of conditions sufficiently so that all possible matching experiments would produce a similar definite outcome.

No real experiment can answer the question that is being asked. The correct answer is: "it depends".

Does that mean that we can not assume that the static friction will be at the limiting value?
Thus, as the friction can take any value, the reaction from the wall is indeterminate?
Can we not rig up an experiment using different surfaces with different coefficients of friction, and measure the reaction from the wall, keeping the surface horizontal and the wall vertical?

It means that the system, as described, has more unknowns than equations. To solve it you need to get more equations. The additional equations come from the details of the shape and material properties.

Until you push the block there is no reaction force from the wall, or from friction. So imagine you start pushing at t=0.... Which kicks in first? The reaction force from the wall or the friction? It can't be determined.